Simple linear functional Errors–In–Variables models with correlated errors

Recognition that measurement error in the independent (predictor) variable of data used for regression modeling necessitates special methods of model fitting dates back to the late 1800s. In the ensuing 100 years attention was devoted almost entirely to measurement error in linear regression models, at which time advances in computing opened the door to the nonlinear statistical modeling generally, and to the study of measurement error in nonlinear regression modeling in particular. This article contains an introduction to the problem of measurement error in regression using linear regression measurement error models to illustrate key ideas. It includes a discussion of the types of measurement error commonly studied (classical and Berkson, differential and nondifferential) and how those types arise in practice. A summary of the consequences of ignoring measurement error when analyzing data measured with error is presented, with detailed results given for simple and multiple linear measurement error models. Pointers to the literature on both linear and nonlinear measurement error models are provided.

In this paper, we introduce the mixed ridge estimator (MRE) in linear measurement error models with stochastic linear restrictions and present the method of weighted mixed ridge estimation, which permits to assign possibly unequal weights to the prior information in relation to the sample information. The performance of the weighted mixed ridge estimator (WMRE) against the weighted mixed estimator (WME) is examined in terms of the mean squared error matrix (MSEM) of estimators. Finally, a simulation study and a numerical example are also given to show the theoretical results.

Linear kinematics model based on POE formula has been widely applied to the error compensation of robot. In this paper, we employed a calibration model in the Local POE frame where all the joint axes are repressed in their respective local frames. The major advantage of this formula is that the local coordinate frames can be arbitrarily assigned onto their corresponding links. For analyzing the applicability of linear model, this paper builds envelope model of error distribution using linear and second-order error model. Meanwhile, this paper analyzes Kuka Youbot robot's error distribution through the established model. The results show that the second-order model could be neglected in comparison with first-order model, when nominal error values of the joint angles are less than 0.1 degree. Hence, linear error model is enough precise in most real applicable environments.

We describe two approaches to instrumental variable estimation in binary regression measurement error models. The methods entail constructing approximate mean models for the binary response as a function of the measured predictor, the instrument, and any covariates in the model. Estimates are obtained by exploiting relationships between regression parameters, just as in linear instrumental variable estimation. In the course of deriving the approximate mean models, we obtain an alternative characterization of instrumental variable estimation in linear measurement error models.

We compare single-valued forecasts from a consensus of numerical weather prediction models to forecasts from a single model across a range of user decision thresholds and sensitivities, using the relative economic value framework, and present this comparison in a new graphical format. With the help of a simple linear error model, we obtain theoretical results and perform synthetic calculations to gain insights into how the results relate to the characteristics of the different forecast systems. We find that multimodel consensus forecasts are more beneficial for users interested in decisions near the climatological mean, due to their reduced spread of errors compared to the constituent models. Single model forecasts may present greater benefit for users sensitive to extreme events if the forecasts have smaller conditional biases than the consensus forecasts and hence better resolution of such events. The results support use of consensus averaging approaches for single-valued forecast services in typical conditions. However, it is hard to cater for all user sensitivities in more extreme conditions. This underscores the importance of providing probability-based services for unusual conditions.

"Prediction and Improved Estimation in Linear Models." Journal of the Operational Research Society, 30(1), pp. 88–89

The problem of optimal joint detection and estimation in linear models with Gaussian noise is studied. A simple closed-form expression for the joint posterior distribution of the (multiple) hypotheses and the states is derived. The expression crystalizes the dependence of the optimal detector on the state estimates. The joint posterior distribution characterizes the beliefs (“soft information”) about the hypotheses and the values of the states. Furthermore, it is a sufficient statistic for jointly detecting multiple hypotheses and estimating the states. The developed expressions give us a unified framework for joint detection and estimation under all performance criteria.

We derive a property of real sequences which can be used to provide a natural sufficient condition for the consistency of the least squares estimators of slope and intercept for a simple linear regression models.

This paper deals with parameter estimation in the linear regression model and an almost unbiased two-parameter estimator is introduced. The performance of this new estimator over the ordinary least-squares estimator and the two-parameter estimator [M.R. Özkale and S. Kaçiranlar, The restricted and unrestricted two-parameter estimator, Comm. Statist. Theory Methods 36 (2007), pp. 2707–2725] in terms of scalar mean-squared error criterion is investigated and a simulation study is done.

This paper deals with asymptotic properties of the Least-Square Estimator and of two-stage estimators in multiariate linear models where the case of singular experimental matrices is included. Fristly we prove the asymptotic optimality of the best linear unbiased estimator (BLUE) when the convariance matrix is assumed to be known in a partial class of linear asymptotically unbiased and consistent sequences of estimators. Comparison of two sequences is made by the help of limits of covariance matrices. Secondly conditions are given under which sequence of Blue's. Furthermore there are a lot of cases in which the ineffienent ordinary Least-Square Estimator (OLSE) becomes asymptotically an efficient estimator when the sample size increases. General conditions are obtained under which this fact holds. Certain examples are discussed for illustration.

Simple and multiple linear regression models are considered between variables whose “values” are convex compact random sets in \({\mathbb{R}^p}\) , (that is, hypercubes, spheres, and so on). We analyze such models within a set-arithmetic approach. Contrary to what happens for random variables, the least squares optimal solutions for the basic affine transformation model do not produce suitable estimates for the linear regression model. First, we derive least squares estimators for the simple linear regression model and examine them from a theoretical perspective. Moreover, the multiple linear regression model is dealt with and a stepwise algorithm is developed in order to find the estimates in this case. The particular problem of the linear regression with interval-valued data is also considered and illustrated by means of a real-life example.

A well-known problem is that ordinary least squares estimation of the parameters in the usual linear model can be highly ineficient when the error term has a heavy-tailed distribution. Inefficiency is also associated with situations where the error term is heteroscedastic, and standard confidence intervals can have probability coverage substantially different from the nominal level. This paper compares the small-sample efficiency of six methods that address this problem, three of which model the variance heterogeneity nonparametrically. Three methods were found to be relatively ineffective, but the other three perform relatively well. One of the six (M-regression with a Huber φ function and Schweppe weights) was found to have the highest efficiency for most of the situations considered in the simulations, but there might be situations where one of two other methods gives better results. One of these is a new method that uses a running interval smoother to estimate the optimal weights in weighted least squares, and the other is a method recently proposed by Cohen, Dalal, and Tukey. Computing a confidence interval for the slope using a bootstrap technique is also considered.

In this article we describe new predictors under multicollinearity situation in the partially linear mixed measurement error models. In order to achieve this aim, we refer to some preliminary information and use it in order to suggest the modified Kernel ridge predictors in the partially linear mixed measurement error models. In addition, we also attain some mean square error comparisons between our new described modified Kernel ridge predictors and predictors previously described in literature for the partially linear mixed measurement error model. In conclusion, the article showcases real data analysis and a simulation study to illusrate our theoretical findings.

Motivated by an example in nutritional epidemiology, we investigate some design and analysis aspects of linear measurement error models with missing surrogate data. The specific problem investigated consists of an initial large sample in which the response (a food frequency questionnaire, FFQ) is observed and then a smaller calibration study in which replicates of the error prone predictor are observed (food records or recalls, FR). The difference between our analysis and most of the measurement error model literature is that, in our study, the selection into the calibration study can depend on the value of the response. Rationale for this type of design is given. Two major problems are investigated. In the design of a calibration study, one has the option of larger sample sizes and fewer replicates or smaller sample sizes and more replicates. Somewhat surprisingly, neither strategy is uniformly preferable in cases of practical interest. The answers depend on the instrument used (recalls or records) and the parameters of interest. The second problem investigated is one of analysis. In the usual linear model with no missing data, method of moments estimates and normal-theory maximum likelihood estimates are approximately equivalent, with the former method in most use because it can be calculated easily and explicitly. Both estimates are valid without any distributional assumptions. In contrast, in the missing data problem under consideration, only the moments estimate is distribution-free, but the maximum likelihood estimate has at least 50% greater precision in practical situations when normality obtains. Implications for the design of nutritional calibration studies are discussed.

This paper explores in depth the nature of the conditional moment restrictions implied by log-linear intertemporal capital asset pricing models (ICAPMs) and shows that the generalized instrumental variables (GMM) estimators of these models (as typically implemented in practice) are inefficient.The moment conditions in the presence of temporally aggregated consumption are derived for two log-linear ICAPMs.The first is a continuous time model in which agents maximize expected utility.In the context of this model, we show that there are important asymmetries between the implied moment conditions for infinitely and finitely-lived securities.The second model assumes that agents maximize non-expected utility, and leads to a very similar econometric relation for the return on the wealth portfolio.Then we describe the efficiency bound (greatest lower bound for the asymptotic variances) of the CNN estimators of the preference parameters in these models.In addition, we calculate the efficient CNN estimators that attain this bound.Finally, we assess the gains in precision from using this optimal CNN estimator relative to the commonly used inefficient CMN estimators.